The function may be called by the names: s14bac, nag_specfun_gamma_incomplete or nag_incomplete_gamma.
3Description
s14bac evaluates the incomplete gamma functions in the normalized form
with and , to a user-specified accuracy. With this normalization, .
Several methods are used to evaluate the functions depending on the arguments and , the methods including Taylor expansion for , Legendre's continued fraction for , and power series for . When both and are large, and , the uniform asymptotic expansion of Temme (1987) is employed for greater efficiency – specifically, this expansion is used when and .
Once either or is computed, the other is obtained by subtraction from . In order to avoid loss of relative precision in this subtraction, the smaller of and is computed first.
This function is derived from the function GAM in Gautschi (1979b).
4References
Gautschi W (1979a) A computational procedure for incomplete gamma functions ACM Trans. Math. Software5 466–481
Temme N M (1987) On the computation of the incomplete gamma functions for large values of the parameters Algorithms for Approximation (eds J C Mason and M G Cox) Oxford University Press
5Arguments
1: – doubleInput
On entry: the argument of the functions.
Constraint:
.
2: – doubleInput
On entry: the argument of the functions.
Constraint:
.
3: – doubleInput
On entry: the relative accuracy required by you in the results. If s14bac is entered with tol greater than or less than machine precision, then the value of machine precision is used instead.
4: – double *Output
5: – double *Output
On exit: the values of the functions and respectively.
6: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALG_NOT_CONV
Algorithm fails to terminate in iterations.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_LE
On entry, .
Constraint: .
NE_REAL_ARG_LT
On entry, .
Constraint: .
7Accuracy
There are rare occasions when the relative accuracy attained is somewhat less than that specified by argument tol. However, the error should never exceed more than one or two decimal places. Note also that there is a limit of decimal places on the achievable accuracy, because constants in the function are given to this precision.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
s14bac is not threaded in any implementation.
9Further Comments
The time taken for a call of s14bac depends on the precision requested through tol, and also varies slightly with the input arguments and .
10Example
This example reads values of the argument and from a file, evaluates the function and prints the results.